An example of an exceptional (-2n-1,n)-curve in an algebraic 3-fold
Let X be a complex nonsingular algebraic threefold. A nonsingular curve \(C\subset X\) is called exceptional if there exists a morphism \(f:X\to Y\) onto a variety Y, contracting C to a point and inducing an isomorphism \(X\setminus C\simeq Y\setminus f(C)\). If \(C\simeq {\mathbb{P}}^ 1\), its normal bundle can be presented as \(N_{C| X}={\mathcal O}_ C(a)\oplus {\mathcal O}_ C(b)\) with \(a\leq b\) and then C is called an exceptional (a,b)-curve. Previous results of the author [cf. Am. J. Math. 113, No.5, 949-961 (1991)] and of \textit{N. Nakayama} [J. Fac. Sci. Univ. Tokyo, Sect. I A 37, No.2, 511-525 (1990)] show that (a,b) must satisfy the conditions \(a+2b\leq -1\), (a,b)\(\neq (-1,0)\) and that for every pair (a,b) in the range \(a+2b\leq -2\) there exists an exceptional (a,b)-curve. In the paper under review the author constructs an exceptional (-2n-1,n)-curve for any \(n>0\). Before, only exceptional (-3,1)-curves were known on the line \(a+2b=-1\).
scientific article; zbMATH DE number 4204549